Editing Fredholm operator (section)

{{Short description|Part of Fredholm theories in integral equations}}
{{main|Fredholm theory}}

In [[mathematics]], '''Fredholm operators''' are certain [[Operator (mathematics)|operators]] that arise in the [[Fredholm theory]] of [[integral equation]]s. They are named in honour of [[Erik Ivar Fredholm]]. By definition, a Fredholm operator is a [[bounded linear operator]] ''T''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''Y'' between two [[Banach space]]s with finite-dimensional [[kernel (algebra)|kernel]] <math>\ker T</math> and finite-dimensional (algebraic) [[cokernel]] <math>\operatorname{coker}T = Y/\operatorname{ran}T</math>, and with closed [[range of a function|range]] <math>\operatorname{ran}T</math>. The last condition is actually redundant.<ref>{{cite book
| last1=Abramovich | first1=Yuri A.
| last2=Aliprantis | first2=Charalambos D.
| title=An Invitation to Operator Theory
| publisher=American Mathematical Society
| series=Graduate Studies in Mathematics
| volume=50
| date=2002
| isbn=978-0-8218-2146-6
| page=156}}</ref>

The ''[[Linear transform#Index|index]]'' of a Fredholm operator is the integer

:<math> \operatorname{ind}T := \dim \ker T - \operatorname{codim}\operatorname{ran}T </math>

or in other words,

:<math> \operatorname{ind}T := \dim \ker T - \operatorname{dim}\operatorname{coker}T.</math>